Down by six is literally that — they were six runs (points) behind. Six is a "magic" number, because that's what you score for knocking the ball out of the park (so the cricket equivalent to hitting a home run).

Yup, batter runs towards the bowler (and the "inactive" batter runs the opposite direction).

In baseball terms, a cricket run is more or less equivalent to running a single base (the bowler is 22 yards away from the batter, which is more than the distance from the pitcher's mound to the home plate, but less than the distance from home to first base). Just like you can run multiple bases in baseball, you can do multiple runs in cricket. From a scoring point of view, you're effectively scoring how many bases you ran, so a baseball run is roughly equivalent to four cricket runs.

Scoring 100+ runs is called "a century", and it's pretty impressive, but, because you keep batting until the bowling team sends you out, you can just keep scoring all day long if you have the endurance for it. Baseball doesn't have a mechanism for a single batter to hit multiple back-to-back home runs.

There's a rope around the field, if the ball goes over it without hitting the ground (like a home run) it counts for 6. If if it did hit the ground it counts for 4.

They can run more than one (get to the other side, turn around, run back, etc) but the chance of the wicket you're running to being hit with the ball (so you're out) becomes larger so they usually manage 1, sometimes 2 or even 3. And both batters have to run the same amount.

If the number of runs is even, they end up on the same side as they started from.